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Time delay profoundly impacts reaction-diffusion systems, which has been considered in many areas, especially infectious diseases, neurodynamics, and chemistry. This paper aims to investigate the pattern dynamics of the reaction-diffusion model with time delay. We obtain the condition in which the system induced the Hopf bifurcation and Turing instability as the parameter of the diffusion term and time delay changed. Meanwhile, the amplitude equation of the reaction-diffusion system with time delay is also derived based on the Friedholm solvability condition and the multi-scale analysis method near the critical point of phase transition. We discussed the stability of the amplitude equation. Theoretical results demonstrate that the delay can induce rich pattern dynamics in the Brusselator reaction-diffusion system, such as strip and hexagonal patterns. It is evident that time delay causes steady-state changes in the spatial pattern under certain conditions but does not cause changes in pattern selection under certain conditions. However, diffusion and delayed feedback affect pattern formation and pattern selection. This paper provides a feasible method to study reaction-diffusion systems with time delay and the development of the amplitude equation. The numerical simulation well verifies and supports the theoretical results. [ABSTRACT FROM AUTHOR]

The epidemic often spreads along social networks and shows the effect of memorability on the outbreak. But the dynamic mechanism remains to be illustrated in the fractional-order epidemic system with a network. In this paper, Turing instability induced by the network and the memorability of the epidemic are investigated in a fractional-order epidemic model. A method is proposed to analyze the stability of the fractional-order model with a network through the Laplace transform. Meanwhile, the conditions of Turing instability and Hopf bifurcation are obtained to discuss the role of fractional order in the pattern selection and the Hopf bifurcation point. These results prove the fractional-order epidemic model may describe dynamical behavior more accurately than the integer epidemic model, which provides the bridge between Turing instability and the outbreak of infectious diseases. Also, the early warning area is discussed, which can be treated as a controlled area to avoid the spread of infectious diseases. Finally, the numerical simulation of the fractional-order system verifies the academic results is qualitatively consistent with the instances of COVID-19. [ABSTRACT FROM AUTHOR]

Neuron modelling helps to understand the brain behavior through the interaction between neurons, but its mechanism remains unclear. In this paper, the spatiotemporal patterns is investigated in a general networked Hindmarsh-Rose (HR) model. The stability of the network-organized system without delay is analyzed to show the effect of the network on Turing instability through the Hurwitz criterion, and the conditions of Turing instability are obtained. Once the analysis of the zero-delayed system is completed, the critical value of the delay is derived to illustrate the profound impact of the given network on the collected behaviors. It is found that the difference between the collected current and the outgoing current plays a crucial role in neuronal activity, which can be used to explain the generation mechanism of the short-term memory. Finally, the numerical simulation is presented to verify the proposed theoretical results. [ABSTRACT FROM AUTHOR]

Turing instability is a prominent feature of reaction–diffusion systems, which is widely investigated in many fields, such as ecology, neurobiology, chemistry. However, although the inhomogeneous diffusion between prey and predators exist in their network space, there are few considerations on how network diffusion affects the stability of prey–predator models. Therefore, in this paper we study the pattern dynamics of a modified reaction–diffusion Holling–Tanner prey–predator model over a random network. Specifically, we study the relationship between the node degrees of the random network and the eigenvalues of the network Laplacian matrix. Then, we obtain conditions under which the network system instability, Hopf bifurcation as well as Turing bifurcation occur. Also, we find an approximate Turing instability region of the diffusion coefficient and the connection probability of the network. Finally, we apply the mean-field approximation theory with numerical simulation to confirm the correctness of our results. The instability region indicates the random migration of the prey and predators among different communities. [ABSTRACT FROM AUTHOR]

P53 plays a vital role in DNA repair, and several mathematical models of the p53-Mdm2 feedback loop were used to explain the biological mechanism. In this paper, a p53-Mdm2 model described by a delay reaction–diffusion equation is studied both analytically and numerically. This research aims to provide an understanding of the impact of delay and sustained pressure on the p53-Mdm2 dynamics and tries to explain some biological mechanism. It is found that the type of pattern formation is affected by Hopf bifurcation. Also, the amplitude equation in delay diffusive system is derived and it is shown that sustained stress plays an essential role in the function of p53. Finally, simulation is used to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Somitogenesis, the sequential formation of a periodic pattern along the anteroposterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this paper, we investigate the effect of diffusion on pattern formation use a modified two dimensional model which can be used to explain somitogenesis during embryonic development. This model is suitable for exploring a design space of somitogenesis and can explain many aspects of somitogenesis that previous models cannot. In the present paper, by analyzing the local linear stability of the equation, we acquired the conditions of Hopf bifurcation and Turing bifurcation. In addition, the amplitude equation near the Turing bifurcation point is obtained by using the methods of multi-scale expansion and symmetry analysis. By analyzing the stability of the amplitude equation, we know that there are various complex phenomena, including Spot pattern, mixture of spot–stripe patterns and labyrinthine. Finally, numerical simulation are given to verify the correctness of our theoretical results. Somitogenesis occupies an important position in the process of biological development, and as a pattern process can be used to investigate many aspects of embryogenesis. Therefore, our study helps greatly to cell differentiation, gene expression and embryonic development. What is more, it is of great significance for the diagnosis and treatment of human diseases to study the related knowledge of model biology. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the effect of diffusion, which is ubiquitous in nature, on the immune system using a reaction-diffusion model in order to understand the dynamical behavior of complex patterns and control the dynamics of different patterns. Through control theory and linear stability analysis of local equilibrium, we obtain the optimal condition under which the system loses stability and a Turing pattern occurs. By combining mathematical analysis and numerical simulation, we show the possible patterns and how these patterns evolve. In addition, we establish a bridge between the complex patterns and the biological mechanism using the results from a previous study in Nature Cell Biology. The results in this paper can help us better understand the biological significance of the immune system. [ABSTRACT FROM AUTHOR]

The diffusion of the susceptible and infected is a vital factor in spreading infectious diseases. However, the previous SIR networks cannot explain the dynamical mechanism of periodic behavior and endemic diseases. Here, we incorporate the diffusion and network effect into the SIR model and describes the mechanism of periodic behavior and endemic diseases through wavenumber and saddle–node bifurcation. We also introduce the standard network structured entropy (NSE) and demonstrate diffusion effect could induce the saddle–node bifurcation and Turing instability. Then we reveal the mechanism of the periodic outbreak and endemic diseases by the mean-field method. We provide the Turing instability condition through wavenumber in this network-organized SIR model. In the end, the data from COVID-19 authenticated the theoretical results. • SIR model with an ER network is constructed to show the effects of the network on Turing instability. • the network structured entropy (NSE) and diffusion could induce the bifurcation and Turing instability. • Mechanism of the periodic outbreak and endemic diseases is revealed. • Turing instability condition was given and verified through wavenumber. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effect of diffusion on pattern formation in FitzHugh–Nagumo model. Through the linear stability analysis of local equilibrium we obtain the condition how the Turing bifurcation, Hopf bifurcation and the oscillatory instability boundaries arise. By using the method of the weak nonlinear multiple scales analysis and Taylor series expansion, we derive the amplitude equations of the stationary patterns. The analysis of amplitude equations shows the occurrence of different complex phenomena, including Turing instability Eckhaus instability and zigzag instability. In addition, we apply this analysis to FitzHugh–Nagumo model and find that this model has very rich dynamical behaviors, such as spotted, stripe and hexagon patterns. Finally, the numerical simulation shows that the analytical results agree with numerical simulation. [ABSTRACT FROM AUTHOR]

• The approximate range of eigenvalue of the network matrix is shown. • The instability condition about diffusion and the connection probability in the network-organized system are derived. • The estimated range of connection probability is also deduced. • Mechanism of spiking of the neuron can be explained by using diffusion network and matrix theory. Although there is general agreement that the network plays an essential role in the biological system, how the connection probability of network affects the natural model(Especially neural network) is poorly understood. In this paper, we show the impact of the network on Turing instability in the FitzHugh-Nagumo(FN) model. Then we obtain the condition of how the Turing bifurcation, saddle-node bifurcation, and Turing instability occur. By using the Gershgorin circle theorem, we investigate the relationship between degree and eigenvalue of the network matrix, and obtain the approximate range of eigenvalue of the network matrix. Also, We derive the instability condition about diffusion and the connection probability in the network-organized system. And then we obtain the estimated range of connection probability. Meanwhile we apply these results to explaining the spiking of neuron and find this system has rich dynamics behavior. Finally, the numerical simulation verifies our analytical results. [ABSTRACT FROM AUTHOR]